Aircraft: COMMERCIAL FIXED WING
Phase of flight: Takeoff / Launch
Anomaly: Conflict Ground Conflict; Critical; Deviation / Discrepancy - Procedural Clearance; Ground Incursion Runway

Aircraft Y had just landed and was instructed to hold short of H9 on H to protect for other arrivals that might exit at H9. When I was ready; I taxied Aircraft Y up to hold short of 25R at l. However; Aircraft Y didn't respond. I called him 3 times with no response. I cleared Aircraft X for takeoff on 25R. My assist asked ground control to see if Aircraft Y had switched frequencies and was just about to call approach to see if Aircraft Y had got back to that frequency when we saw Aircraft Y taxiing forward. I called Aircraft Y and verified he would hold short of Runway 25R which he read back with his call sign. Shortly after my assist said 'He's not going to stop!' Momentarily I didn't believe it but quickly went to Aircraft Y and told him to stop. I then immediately cancelled Aircraft X takeoff clearance. Aircraft Y did stop with their nose about at the runway edge. Aircraft X was about at B2 when I cancelled their clearance. Aircraft X was already stopping before ASDE-X (Airport Surface Detection Equipment) alarmed. Aircraft X taxied back for departure and I crossed Aircraft Y. Maybe I shouldn't have departed Aircraft X with Aircraft Y not responding. Aircraft Y was holding on taxiway H short of H9. Aircraft X was told to Line up and wait Runway 25R. Aircraft Y was then told to turn left at H and Hold Short Runway 25R. There was no response from Aircraft Y. The instruction was repeated again to Aircraft Y but still no response. I on Local Assist went to Ground Control to see if they had Aircraft Y on frequency which they did not. I noticed Aircraft Y start moving. Local Control reached out again with and repeated Hold Short Runway 25R. This time Aircraft Y responded and gave a good read back of Hold Short Runway 25R.Aircraft X was cleared for takeoff and started the departure roll. I first noticed that as Aircraft Y was turning at taxiway Lima approaching the hold bars it did not look like they were stopping; and verbal said that. I immediately said again that Aircraft Y is not stopping. Local Control told Aircraft Y to hold position and canceled the takeoff clearance for Aircraft X as they were between B3 and G. Two seconds later the Airport Surface Detection Equipment alarm went off for the occupied runway alert. Aircraft X exited Runway 25R at B4 and taxied back for departure. Aircraft Y continued the cross and went to parking.I don't know what to recommend to prevent this from happening again as all the protocols were followed with correct read backs. I do not know if the Runway Stop Lights were working properly. I know there were issues with that system in the past with the red lights turning off early. But since it was daytime and Lima is not in a direct line of sight for those lights; I cannot know if the system was functioning properly. LAX Tower Local Controller and the Local Assist Controller reported an aircraft which had been instructed to hold short of the runway taxied on to the runway at the same time a departure was beginning their takeoff roll.